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A conservation law\index{conservation law} is a balance that can be applied to every connex domain strictly interior to the considered system and that is followed in its movement. such a law can be written:

Symbol ddt{\displaystyle {\frac {d}{dt}}} represents the particular derivative (see appendix chapretour). Ai{\displaystyle A_{i}} is a scalar or tensorial\footnote{ Ai{\displaystyle A_{i}} is the volumic density of quantity A{\displaystyle {\mathcal {A}}} (mass, momentum, energy ...). The subscript i{\displaystyle i} symbolically designs all the subscripts of the considered tensor. } function of eulerian variables x{\displaystyle x} and t{\displaystyle t}. ai{\displaystyle a_{i}} is volumic density rate provided by the exterior to the system. αij{\displaystyle \alpha _{ij}} is the surfacic density rate of what is lost by the system through surface bording D{\displaystyle D}.

Equation eqcon represents the integral form of a conservation law. To this integral form is associated a local form that is presented now. As recalled in appendix chapretour, we have the following relation: